Continuous-time active complex bandpass (BP) filters are widely used in the integrated receivers. They serve primarily as intermediate-frequency (IF) channel select filters with additional function of providing image rejection. Their ability to reject unwanted image frequencies of the preceding mixer results directly from their non-symmetrical transfer characteristic. Depending on its input signal conditioning a complex BP filter transmits for positive frequencies and rejects all negative frequencies, or vice-versa, it transmits for negative frequencies and blocks all positive frequencies.
The achievable image-rejection ratio (IRR) depends on the matching of on-chip components used in a complex filter. They include resistors, capacitors and transconductors. Also, non-ideal gain of opamps if used for filter synthesis results in IRR degradation. The IRR performance also depends on the choice of synthesis method. Certain methods are more sensitive to component variation than the others. Practically achievable IRR of a complex filter is 30–35 dB. If extreme caution is taken to achieve an excellent component matching, or if a special automatic compensation scheme is applied the IRR can reach 55–60 dB.
Active complex bandpass filters can be realized using two distinct synthesis methods: the active ladder simulation and the direct synthesis. Similarly to the classical passive LC ladder synthesis method, in the active ladder simulation each pole frequency and its quality (Q) factor are defined by all filter components. Contrary, for the direct synthesis method the pole frequency and its Q factor are defined only by the components of one particular filter section. Due to its lower sensitivity to the component value variation, the active ladder simulation method is superior to the direct synthesis method and it is the preferred method for realization of complex active bandpass filters.
Active complex bandpass filters can also be categorized according to the active synthesis method chosen for their implementation. Two different synthesis techniques have been used so far: the active-RC technique used in U.S. Pat. No. 4,914,408 and the gyrator method used in U.S. Pat. No. 6,346,850. It should be noted that both techniques were widely known prior to their application in the above-mentioned Patents. In active-RC method the transfer function is realized using active-RC integrators built with input series resistors and feedback capacitors around operational amplifiers (opamps). The gyrator method uses voltage-controlled current sources and capacitors to realize integrators. The advantage of the gyrator method over the active-RC method stems from its ability to adjust the filter pole frequencies by adjusting the transconductance with a control voltage or current, which is not easily achievable in a fixed R and C arrangement of the active RC filters.
Due to their prime application as channel select filters complex BP filters must demonstrate sharp roll-off outside their pass-bands. In wireless receiver system design their attenuation determines such critical parameters as co-channel and adjacent channel rejection. Steep roll-off is not easily achievable with all-pole transfer functions. Depending on their Order all-pole transfer functions may be quite steep, but as illustrated in FIG. 16, their roll-off is never as steep as that of filters that contain transmission zeros. For all these reasons, in an integrated receiver design there is a strong need for active complex BP filters with transmission zeros. Ladders offer a wide range of transfer functions: both all-pole prototypes and filters with transmission zeros.